Transition-rate matrix

Matrix describing continuous-time Markov chains From Wikipedia, the free encyclopedia

In probability theory, a transition-rate matrix (also known as a Q-matrix,[1] intensity matrix,[2] or infinitesimal generator matrix[3]) is an array of numbers describing the instantaneous rate at which a continuous-time Markov chain transitions between states.

In a transition-rate matrix (sometimes written [4]), element (for ) denotes the rate departing from and arriving in state . The rates , and the diagonal elements are defined such that

,

and therefore the rows of the matrix sum to zero.

Up to a global sign, a large class of examples of such matrices is provided by the Laplacian of a directed, weighted graph. The vertices of the graph correspond to the Markov chain's states.

Properties

Summarize
Perspective

The transition-rate matrix has following properties:[5]

  • There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of is strongly connected.
  • All other eigenvalues fulfill .
  • All eigenvectors with a non-zero eigenvalue fulfill .
  • The Transition-rate matrix satisfies the relation where P(t) is the continuous stochastic matrix.

Example

Summarize
Perspective

An M/M/1 queue, a model which counts the number of jobs in a queueing system with arrivals at rate λ and services at rate μ, has transition-rate matrix

See also

References

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